Solve the following quadratic equations by the method of completing the square:

4x² + 3x + 5 = 0

Now in the above quadratic equation the coefficient of x² is 4. Let us make it unity by dividing the entire quadratic equation by 4.

x² – 3/4x + 5/4 = 0

x² – 3/4x = -5/4

Now by taking half of the coefficient of x and then squaring it and adding on both LHS and RHS sides.

Coefficient of x = 3/4

Half of 3/4 = 3/8

Squaring the half of 3/4 = 9/64

Now the LHS term is a perfect square and can be expressed in the form of (a-b) ² = a² – 2ab + b² where a = x and b = 3/8

On simplifying both RHS and LHS we get an equation of following form,

(x ± A)² = k²

It is observed that the term obtained on RHS is a negative term and taking square root of a negative term will give imaginary roots for the given quadratic equation.

Therefore the given quadratic equation does not has real roots.